Bifurcation of the Equilibrium Point in the Critical Case of Two Pairs of Zero Characteristic Roots

A real autonomous system of four differential equations with a small parameter is considered. It is proved that, under a finite number of explicit conditions on the coefficients of the lower order terms in the expansion of the right-hand sides, its two-dimensional invariant torus bifurcates at infinitesimal frequencies for sufficiently small values of the parameter. Such a system describes, in particular, the oscillations of two weakly coupled oscillators with restoring forces of orders $2n-1$ and $2n+1$.